Masao Aoki
We study Hom 2-functors parameterizing 1-morphisms of algebraic stacks, and prove that it is representable by an algebraic stack under certain conditions, using Artin's criterion. As an application we study Picard 2-functors which parameterizes line bundles on algebraic stacks.
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The purpose of this note is to prove that there is an algebraic stack U parameterizing all curves. The curves that appear in the algebraic stack U are allowed to be arbitrarily singular, non-reduced, disconnected, and reducible. We also prove the boundedness of the open substack of U parameterizing geometrically connected curves with fixed arithmetic genus g and $\leq$ e irreducible components. This is an updated and expanded version of [arXiv:0902.3690v2, Appendix B].
Let π: X -> S be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack, the Hilbert stack, parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.
Existence of moduli spaces for algebraic stacks
本文给出代数栈存在良好模空间的充要条件，证明带 $Θ$-分层的半稳定约化定理，推广 Keel--Mori 定理并应用于构造半稳定 $\mathcal{G}$-丛和阿贝尔范畴对象的紧合模空间。
We show that there is a logarithmic algebraic space parameterizing logarithmic morphisms between fixed logarithmic schemes when those logarithmic schemes satisfy natural hypotheses. As a corollary, we obtain the algebraicity of the stack of stable logarithmic maps without restriction on the logarithmic structure of the target.
We establish Tannaka duality for noetherian algebraic stacks with affine stabilizer groups. Our main application is the existence of Hom-stacks in great generality.
Motivic Homotopy Theory of Algebraic Stacks
本文利用∞-范畴语言和增强运算映射，将动机同伦理论及六函子形式体系从概形推广到一大类代数栈，并证明其满足同伦不变性、局部化和纯性。
On the Motivic Homotopy Type of Algebraic Stacks
本文构造了在Morel-Voevodsky A1-同伦范畴中为局部满射的代数栈的光滑展示，并证明光滑栈的动机具有与光滑概形动机相似的许多性质。
The Morel-Voevodsky Construction over Algebraic Stacks
本文以Morel-Voevodsky精神构造了代数栈的（非）稳定motivic同伦范畴，证明其与Chowdhury-D'Angelo的定义一致，并将Bachmann谱刚性定理、带框motivic同伦范畴及Hoyois重建定理推广至代数栈。
We construct an algebraic homology functor for Artin stacks of finite type over a field, and we develop intersection-theoretic properties.
We prove that if a group scheme of multiplicative type acts on an algebraic stack with affine, finitely presented diagonal then the stack of fixed points is algebraic. For this, we extend two theorems of [SGA3.2] on functors of subgroups of multiplicative type, and functors of homomorphisms from a group of multiplicative type.
We prove the algebraicity of the Hilbert functor, the Hilbert stack, the Quot functor and the stack of coherent sheaves on an algebraic stack X with (quasi-)finite diagonal without any finiteness assumptions on X. We also give similar results for Hom stacks and Weil restrictions.
We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U \subset X$ be an open embedding of smooth Deligne-Mumford stacks such that $D = X-U$ is a normal crossings divisor, then the the simple homotopy type of the boundary complex $Δ(X,D)$ depends only on $U$.
Using the machinery of etale homotopy theory a' la Artin-Mazur we determine the etale homotopy types of moduli stacks over $\bar{\Q}$ parametrizing families of algebraic curves of genus g greater than 1 endowed with an action of a finite group G of automorphisms, which comes with a fixed embedding in the mapping class group, such that in the associated complex analytic situation the action of G is precisely the differentiable action induced by this specified embedding of G in the mapping class group.
On a smooth algebraic curve X with genus greater than 1 we consider a flat principal bundle with a reductive structure group S and a vector bundle associated with it. To this set of information we put in correspondence a pro-algebraic group on whose functional algebra we introduce a mixed Hodge structure. This construction, in fact, works for any smooth algebraic variety X which, considered as an analytic space, has a nonabelian first homotopy group, and the rest are trivial. The Hodge structure defined in this way can be expressed in terms of iterated integrals. Furthermore, considered in the…
Rational Homotopy and Hodge Theory of Moduli Stacks of principal $G$-bundles
本文利用拓扑栈的同伦理论，确定了特殊类型复射影簇上主$G$-丛模栈的有理上同调与Hodge-Tate结构。
We prove that cohomology and base change holds for algebraic stacks, generalizing work of Brochard in the tame case. We also show that Hom-spaces on algebraic stacks are represented by abelian cones, generalizing results of Grothendieck, Brochard, Olsson, Lieblich, and Roth--Starr. To accomplish all of this, we prove that a wide class of Ext-functors in algebraic geometry are coherent (in the sense of M. Auslander).
Cohomological integrality for symmetric quotient stacks
本文建立了对称商栈的上同调完整性猜想的层化版本，通过定义BPS层并证明其纯度，进而证明了Halpern-Leistner关于自对偶余切复形导出栈的Borel-Moore上同调纯度的猜想，并应用于主Higgs丛模栈。
We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem which shows that the weak homotopy type of the mapping stack Map(Y,X) does not change if we replace X by its classifying space, provided that Y is paracompact topological space. As an example, we describe the loop stack of the classifying s…
This is an expended and revised version of the preprint "Schematization of homotopy types". The purpose of this work is to introduce a notion of \emph{affine stacks}, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and $p$-adic homotopy theory. This gives a first solution to A. Grothendieck's \emph{schematization problem} described in \cite{gr}. We also use affine…
This is the first in a series of papers devoted to foundations of topological stacks. We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of topological spaces. In this paper we go as far as introducing the homotopy groups and establishing their basic properties. We also develop a Galois theory of covering spaces for a (locally connected semilocally 1-connected) topological stack. Built into the Galois theory is a method for determining the stacky structure (i.e., inertia groups) of covering stacks. As a consequence, we get for free a chara…
This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category, such as derivations, etale and smooth maps, flat and projective modules, etc. We then use the theory of stacks over model categories introduced in \cite{hagI} in order to define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties. The rest of the paper consists in …
We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.
We prove a conjecture on the relation between dimer models, coamoebas and vanishing cycles for the mirrors of two-dimensional toric Fano stacks of Picard number one. As a corollary, we obtain a torus-equivariant version of homological mirror symmetry for such stacks.
We establish normal form theorems for a large class of singular flat connections on complex manifolds, including connections with logarithmic poles along weighted homogeneous Saito free divisors. As a result, we show that the moduli spaces of such connections admit the structure of algebraic quotient stacks. In order to prove these results, we introduce homogeneous Lie groupoids and study their representation theory. In this direction, we prove two main results: a Jordan-Chevalley decomposition theorem, and a linearization theorem. We give explicit normal forms for several examples of free div…
Generalized local jacobians and commutative group stacks
本文通过更高代数群栈的方法重构了Grothendieck提出的广义局部雅可比构造，并将其推广到任意基上。
We define a closed model category containing the $n$-nerves defined by Tamsamani, and admitting internal $Hom$. This allows us to construct the $n+1$-category $nCAT$ by taking the internal $Hom$ for fibrant objects. We prove a generalized Seifert-Van Kampen theorem for Tamsamani's Poincaré $n$-groupoid of a topological space. We give a still-speculative discussion of $n$-stacks, and similarly of comparison with other possible definitions of $n$-category.
Generalized cohomology theories for algebraic stacks
本文扩展了Voevodsky的稳定motivic同调范畴至scalloped代数堆，展示其具备Grothendieck的六操作公理，为堆上的广义上同调理论提供框架，包括代数K理论和新的例子如motivic上同调和代数 bordism。
Hochschild homology for log schemes
将Hochschild同调和循环同调推广到代数空间到代数栈的态射，进而得到Fontaine-Illusie意义下对数方案的这些同调理论。
We define a tricategory T of length 3 complexes of abelian sheaves, whose hom-bigroupoids consist of weak morphisms of such complexes. We also define a 3-category 2PIC(S) of Picard 2-stacks, whose hom-2-groupoids consist of additive 2-functors. We prove that these categories are triequivalent as tricategories. As a consequence we obtain a generalization of Deligne's analogous result about Picard stacks in SGA4, Exp. XVIII.
We prove some fundamental results like localization, excision, Nisnevich descent and the Mayer-Vietoris property for equivariant regular blow-up for the equivariant K-theory of schemes with an affine group scheme action. We also show that the equivariant K-theory with finite coefficients is invariant under equivariant vector bundle morphisms. We show that the equivariant homotopy K-theory is invariant under equivariant vector bundle morphisms and satisfies all the above properties, including nil-invariance.
Given an algebraic stack $X$, one may compare the derived category of quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring of functions on $X$. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber-exact homology theories. We identify a condition (quasi-affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and van…
We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack $X/G$ and its classical $z$-fixed point stack $π_0(X^z)/G^z$ become equivalent after completion along a semisimple parameter $[z] \in G//G$, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes $\text{Perf}(X/G)$. We then prove an analogue of th…
We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic $τ$-deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over $\mathbf C$ with ind-coherent sheaves (in a slightly non-standard sense) on a certain spectral stack $τ_{\ge 0}(\mathcal M_\mathrm{FG}^\mathrm{or})$. The latter is the connective cover of the non-connective spectral stack $\mathcal M_\mathrm{FG}^\mathrm{or}$, the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chr…
We prove structure theorems for algebraic stacks with a reductive group action and a dense open substack isomorphic to a horospherical homogeneous space, and thereby obtain new examples of algebraic stacks which are global quotient stacks. Our results partially generalize the work of Fantechi-Mann-Nironi and Geraschenko-Satriano for abstract toric stacks.
We show Goodwillie's calculus of functors and $n$-geometric $D^{-}$-stacks share similar features by starting to focus on the convergence of Taylor towers for homotopy functors and the fact that $\mathbb{R} F(A) \cong \text{holim} \mathbb{R} F(A_{\leq n})$ for geometric stacks, where $\{ A_{\leq n} \}$ provides a Postnikov tower of some given $A \in s \text{k-Alg}$. From there we show parallel results, such as similar homotopy fibers of connecting maps in towers, as well as polynomial approximations, pointwise approximations and reconstruction theorems for towers.
These are expanded notes from some talks given during the fall 2002, about ``homotopical algebraic geometry'' (HAG) with special emphasis on its applications to ``derived algebraic geometry'' (DAG) and ``derived deformation theory''. We use the general framework developed in Toen, Vezzosi, ``Homotopical Algebraic Geometry I: Topos theory'', and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of …
We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For $Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence of deformation retracts and attachments of high-dimensional finite disc quotients. We use this to derive more familiar statements about the relative homotopy, homology, and cohomology groups of the pair $(X,Z)$. We also prove some preliminary results suggesting that the Lefschetz Hyperplane Theorem holds for Artin stacks as well. One technical innovation is to rei…
This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts (for part II, see math.AG/0404373). In this first part we investigate a notion of higher topos. For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of \infty-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on s…
Root stack valuative criterion for good moduli spaces
本文在剩余特征的温和假设下，证明了好模空间映射与约化群gerbe的根叠赋值判别法，并给出了其在挠子parahoric扩张、叠有理点、gerbe与齐性空间及纤维丛几何中的应用。
Berkovich $2$-motives and normed ring stacks
本文提出并证明了动机同调理论核范畴与同调平凡的光滑缝合环堆栈的关系，并探讨了其在分析几何中的应用。
In this paper, we give an expository account of the geometric properties of the moduli stack of $G$-bundles. For $G$ an algebraic group over a base field and $X \to S$ a flat, finitely presented, projective morphism of schemes, we give a complete proof that the moduli stack $Bun_G$ is an algebraic stack locally of finite presentation over $S$ with schematic, affine diagonal. In the process, we prove some properties of $BG$ and Hom stacks. We then define a level structure on $Bun_G$ to provide alternative presentations of quasi-compact open substacks. Finally, we prove that $Bun_G$ is smooth ov…
We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there i…
Homotopy Wave Function in Algebraic Geometry
本文提出将simplicial k-algebras上derived stacks的Segal topos的同伦形状作为量子力学波函数概念的高阶同伦推广。
We show that if G is a finite constant group acting on a scheme X such that the order of G is invertible in the residue fields of X, then the G-equivariant motivic stable homotopy category of X is equivalent to the stabilization of the category of motivic G-spaces with finite étale transfers over X at the trivial representation sphere. Along the way we obtain several results of independent interest, among them: we construct and study norms in the motivic homotopy theory of stacks, and we extend the homotopy t-structure to DM-stacks and establish some favorable properties.
We prove that the homotopy algebraic K-theory of tame quasi-DM stacks satisfies cdh-descent. We apply this descent result to prove that if X is a Noetherian tame quasi-DM stack and i < -dim(X), then K_i(X)[1/n] = 0 (resp. K_i(X, Z/n) = 0) provided that n is nilpotent on X (resp. is invertible on X). Our descent and vanishing results apply more generally to certain Artin stacks whose stabilizers are extensions of finite group schemes by group schemes of multiplicative type.
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